[Paper Review] On [L]-homotopy groups

A. Karasev|ArXiv.org|Jan 4, 2000

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper establishes that for any finite CW-complex $ L $ satisfying $[S^n] < [L] \leq [S^{n+1}]$, the $[L]$-homotopy group $\pi_n^{[L]}(S^n)$ is isomorphic to $\mathbb{Z}$. Using extension dimension theory and $[L]$-homotopy equivalence, the authors construct a compactum via an inverse system with $[L]$-resolvable properties and derive a contradiction assuming $\pi_n^{[L]}(S^n) \cong \mathbb{Z}_m$, proving the group must be infinite cyclic.

ABSTRACT

Some properties of [L]-homotopy group for finite complex L are investigated. It is proved that for complex L whose extension type lying between Sn and Sn+1 n-th [L]-homotopy group of Sn is isomorphic to Z.

Motivation & Objective

• To compute $[L]$-homotopy groups of spheres in terms of standard homotopy groups and $L$'s extension type.
• To resolve the open problem of determining $\pi_n^{[L]}(S^n)$ when $[L]$ lies strictly between $[S^n]$ and $[S^{n+1}]$.
• To establish that $\pi_n^{[L]}(S^n) \cong \mathbb{Z}$ for such $L$, extending classical homotopy theory to the $[L]$-homotopy framework.

Proposed method

• Define $[L]$-homotopy via extension properties over spaces $X$ with $\mathop{\rm ed}\nolimits(X) \leq [L]$.
• Use $[L]$-universal $\mathop{\rm ANE}\nolimits([L])$ compacta and approximately $[L]$-soft maps to construct $[L]$-spheres $S^n_{[L]}$.
• Construct an inverse system $\{X_i, p^{i+1}_i\}$ of compact polyhedra with $X_0 = D^{n+1}$, $\mathop{\rm mesh}\nolimits \to 0$, and $[L]$-resolvability.
• Apply the Vietoris-Begle theorem to analyze cohomology isomorphisms $p_0^*: H^k(S^n; \mathbb{Z}_p) \to H^k(A; \mathbb{Z}_p)$ for $k \leq n$.
• Use the exact sequence of the pair $(X, A)$ to analyze $\delta^*_{X,A}$ and derive a contradiction when assuming $\pi_n^{[L]}(S^n) \cong \mathbb{Z}_m$.
• Leverage the fact that $p_0^*(\zeta)$ generates $H^n(A; \mathbb{Z}_p) \cong \mathbb{Z}_p$ and that $h = z_m p_0$ would imply $\delta^*_{X,A}(h^*(\zeta)) = m \delta^*_{X,A}(p_0^*(\zeta)) \neq 0$ if $p > m$, contradicting extension.

Experimental results

Research questions

• RQ1 What is the structure of $\pi_n^{[L]}(S^n)$ when $[L]$ lies strictly between $[S^n]$ and $[S^{n+1}]$?
• RQ2 Can $[L]$-homotopy groups be computed using standard homotopy groups and cohomological data of $L$?
• RQ3 Does the $[L]$-homotopy group $\pi_n^{[L]}(S^n)$ remain isomorphic to $\mathbb{Z}$ under the given extension type constraints?

Key findings

• For any finite CW-complex $L$ satisfying $[S^n] < [L] \leq [S^{n+1}]$, the $[L]$-homotopy group $\pi_n^{[L]}(S^n)$ is isomorphic to $\mathbb{Z}$.
• The proof relies on constructing a compactum $X = \lim S$ from a $[L]$-resolvable inverse system with $X_0 = D^{n+1}$ and $\mathop{\rm mesh}\nolimits \to 0$.
• Cohomological analysis shows $H^n(X; \mathbb{Z}_p) = 0$, and $p_0^*: H^n(S^n; \mathbb{Z}_p) \to H^n(A; \mathbb{Z}_p)$ is an isomorphism, so $p_0^*(\zeta)$ is a generator of order $p$.
• The composition $h = z_m p_0$ cannot extend over $X$ if $p > m$, contradicting the assumption that $\pi_n^{[L]}(S^n) \cong \mathbb{Z}_m$, proving the group must be $\mathbb{Z}$.
• The contradiction arises from $\delta^*_{X,A}(h^*(\zeta)) = m \delta^*_{X,A}(p_0^*(\zeta)) \neq 0$ in $H^{n+1}(X,A; \mathbb{Z}_p)$, but extension would require it to vanish.